# gsorth {heplots}

Orthogonalize successive columns of a data frame or matrix
Package:
heplots
Version:
1.0-11

### Description

`gsorth` uses sequential, orthogonal projections, as in the Gram-Schmidt method, to transform a matrix or numeric columns of a data frame into an uncorrelated set, possibly retaining the same column means and standard deviations as the original.

In statistical applications, interpretation depends on the `order` of the variables orthogonalized. In multivariate linear models, orthogonalizing the response, Y variables provides the equivalent of step-down tests, where Y1 is tested alone, and then Y2.1, Y3.12, etc. can be tested to determine their additional contributions over the previous response variables.

Similarly, orthogonalizing the model X variables provides the equivalent of Type I tests, such as provided by `anova`.

### Usage

```gsorth(y, order, recenter = TRUE, rescale = TRUE, adjnames = TRUE)
```

### Arguments

y
A numeric data frame or matrix
order
An integer vector specifying the order of and/or a subset of the columns of `y` to be orthogonalized. If missing, `order=1:p` where `p=ncol(y)`.
recenter
If `TRUE`, the result has same column means as original; else means = 0 for cols `2:p`.
rescale
If `TRUE`, the result has same column standard deviations as original; else sd = residual variance for cols `2:p`
If `TRUE`, the column names of the result are adjusted to the form Y1, Y2.1, Y3.12, by adding the suffixes '.1', '.12', etc. to the original column names.

### Details

The method is equivalent to setting each of columns `2:p` to the residuals from a linear regression of that column on all prior columns, i.e.,

`z[,j] <- resid( lm( z[,j] ~ as.matrix(z[,1:(j-1)]), data=z) )`

However, for accuracy and speed the transformation is carried out using the QR decomposition.

### Values

Returns a matrix or data frame with uncorrelated columns. Row and column names are copied to the result.

`qr`,

### Examples

```GSiris <- gsorth(iris[,1:4])
GSiris <- gsorth(iris, order=1:4)   # same, using order
str(GSiris)
zapsmall(cor(GSiris))
colMeans(GSiris)
# sd(GSiris) -- sd(<matrix>) now deprecated
apply(GSiris, 2, sd)

# orthogonalize Y side
GSiris <- data.frame(gsorth(iris[,1:4]), Species=iris\$Species)
iris.mod1 <- lm(as.matrix(GSiris[,1:4]) ~ Species, data=GSiris)
Anova(iris.mod1)

# orthogonalize X side
rohwer.mod <- lm(cbind(SAT, PPVT, Raven) ~ n + s + ns + na + ss, data=Rohwer)
Anova(rohwer.mod)

# type I tests for Rohwer data
Rohwer.orth <- cbind(Rohwer[,1:5], gsorth(Rohwer[, c("n", "s", "ns", "na", "ss")], adjnames=FALSE))

rohwer.mod1 <- lm(cbind(SAT, PPVT, Raven) ~ n + s + ns + na + ss, data=Rohwer.orth)
Anova(rohwer.mod1)
# compare with anova()
anova(rohwer.mod1)

# compare heplots for original Xs and orthogonalized, Type I
heplot(rohwer.mod)
heplot(rohwer.mod1)```

### Author(s)

Michael Friendly

Documentation reproduced from package heplots, version 1.0-11. License: GPL (>= 2)